Spectral cluster bounds for orthonormal functions on manifolds with nonsmooth metrics

TL;DR

This paper derives $L^q$ bounds for spectral clusters of orthonormal functions on compact manifolds with nonsmooth metrics, extending previous results to metrics with limited regularity.

Contribution

It establishes spectral cluster bounds for orthonormal functions on manifolds with $C^s$ metrics, broadening the scope of spectral analysis in less regular geometric settings.

Findings

Spectral cluster bounds hold for $C^s$ metrics with $0 \\leq s \\leq 2$.

Results extend classical bounds to nonsmooth metric contexts.

Provides tools for spectral analysis on manifolds with limited regularity.

Abstract

We establish $L^q$ spectral cluster bounds for families of orthonormal functions associated to the Laplace-Beltrami operator on a compact Riemannian manifold. The metric is only assumed to be of class $C^s$, where $0\leq s\leq 2$.